Reusable garbled circuits and succinct functional encryption

Garbled circuits, introduced by Yao in the mid 80s, allow computing a function f on an input x without leaking anything about f or x besides f(x). Garbled circuits found numerous applications, but every known construction suffers from one limitation: it offers no security if used on multiple inputs x. In this paper, we construct for the first time reusable garbled circuits. The key building block is a new succinct single-key functional encryption scheme. Functional encryption is an ambitious primitive: given an encryption Enc(x) of a value x, and a secret key sk_f for a function f, anyone can compute f(x) without learning any other information about x. We construct, for the first time, a succinct functional encryption scheme for {\em any} polynomial-time function f where succinctness means that the ciphertext size does not grow with the size of the circuit for f, but only with its depth. The security of our construction is based on the intractability of the Learning with Errors (LWE) problem and holds as long as an adversary has access to a single key sk_f (or even an a priori bounded number of keys for different functions). Building on our succinct single-key functional encryption scheme, we show several new applications in addition to reusable garbled circuits, such as a paradigm for general function obfuscation which we call token-based obfuscation, homomorphic encryption for a class of Turing machines where the evaluation runs in input-specific time rather than worst-case time, and a scheme for delegating computation which is publicly verifiable and maintains the privacy of the computation.Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant)United States. Defense Advanced Research Projects Agency (DARPA award FA8750-11-2-0225)United States. Defense Advanced Research Projects Agency (DARPA award N66001-10-2-4089)National Science Foundation (U.S.) (NSF award CNS-1053143)National Science Foundation (U.S.) (NSF award IIS-1065219)Google (Firm

Indistinguishability Obfuscation for Turing Machines: Constant Overhead and Amortization

We study the asymptotic efficiency of indistinguishability obfuscation (iO) on two fronts: - Obfuscation size: Present constructions of indistinguishability obfuscation (iO) create obfuscated programs where the size of the obfuscated program is at least a multiplicative factor of security parameter larger than the size of the original program. In this work, we construct the first iO scheme for (bounded-input) Turing machines that achieves only a constant multiplicative overhead in size. The constant in our scheme is, in fact, 2. - Amortization: Suppose we want to obfuscate an arbitrary polynomial number of (bounded-input) Turing machines M_1,...,M_n. We ask whether it is possible to obfuscate M_1,...,M_n using a single application of an iO scheme for a circuit family where the size of any circuit is independent of n as well the size of any Turing machine M_i. In this work, we resolve this question in the affirmative, obtaining a new bootstrapping theorem for obfuscating arbitrarily many Turing machines. Our results rely on the existence of sub-exponentially secure iO for circuits and re-randomizable encryption schemes. In order to obtain these results, we develop a new template for obfuscating Turing machines that is of independent interest and has recently found application in subsequent work on patchable obfuscation [Ananth et al, EUROCRYPT\u2717]

Patchable Indistinguishability Obfuscation: iO for Evolving Software

In this work, we introduce patchable indistinguishability obfuscation: our notion adapts the notion of indistinguishability obfuscation (iO) to a very general setting where obfuscated software evolves over time. We model this broadly by considering software patches P as arbitrary Turing Machines that take as input the description of a Turing Machine M, and output a new Turing Machine description M\u27 = P(M). Thus, a short patch P can cause changes everywhere in the description of M and can even cause the description length of the machine to increase by an arbitrary polynomial amount. We further consider multi-program patchable indistinguishability obfuscation where a patch is applied not just to a single machine M, but to an unbounded set of machines M_1,..., M_n to yield P(M_1),.., P(M_n). We consider both single-program and multi-program patchable indistinguishability obfuscation in a setting where there are an unbounded number of patches that can be adaptively chosen by an adversary. We show that sub-exponentially secure iO for circuits and sub-exponentially secure re-randomizable encryption schemes imply single-program patchable indistinguishability obfuscation; and we show that sub-exponentially secure iO for circuits and sub-exponentially secure DDH imply multi-program patchable indistinguishability obfuscation. At the our heart of results is a new notion of splittable iO that allows us to transform any iO scheme into a patchable one. Finally, we exhibit some simple applications of patchable indistinguishability obfuscation, to demonstrate how these concepts can be applied

Time-Lock Puzzles from Randomized Encodings

Time-lock puzzles are a mechanism for sending messages "to the future". A sender can quickly generate a puzzle with a solution s that remains hidden until a moderately large amount of time t has elapsed. The solution s should be hidden from any adversary that runs in time significantly less than t, including resourceful parallel adversaries with polynomially many processors. While the notion of time-lock puzzles has been around for 22 years, there has only been a single candidate proposed. Fifteen years ago, Rivest, Shamir and Wagner suggested a beautiful candidate time-lock puzzle based on the assumption that exponentiation modulo an RSA integer is an "inherently sequential" computation. We show that various flavors of randomized encodings give rise to time-lock puzzles of varying strengths, whose security can be shown assuming the mere existence of non-parallelizing languages, which are languages that require circuits of depth at least t to decide, in the worst-case. The existence of such languages is necessary for the existence of time-lock puzzles. We instantiate the construction with different randomized encodings from the literature, where increasingly better efficiency is obtained based on increasingly stronger cryptographic assumptions, ranging from one-way functions to indistinguishability obfuscation. We also observe that time-lock puzzles imply one-way functions, and thus the reliance on some cryptographic assumption is necessary. Finally, generalizing the above, we construct other types of puzzles such as proofs of work from randomized encodings and a suitable worst-case hardness assumption (that is necessary for such puzzles to exist)

Garbling Schemes and Applications

The topic of this thesis is garbling schemes and their applications. A garbling scheme is a set of algorithms for realizing secure two-party computation. A party called a client possesses a private algorithm as well as a private input and would like to compute the algorithm with this input. However, the client might not have enough computational resources to evaluate the function with the input on his own. The client outsources the computation to another party, called an evaluator. Since the client wants to protect the algorithm and the input, he cannot just send the algorithm and the input to the evaluator. With a garbling scheme, the client can protect the privacy of the algorithm, the input and possibly also the privacy of the output. The increase in network-based applications has arisen concerns about the privacy of user data. Therefore, privacy-preserving or privacy-enhancing techniques have gained interest in recent research. Garbling schemes seem to be an ideal solution for privacy-preserving applications. First of all, secure garbling schemes hide the algorithm and its input. Secondly, garbling schemes are known to have efficient implementations. In this thesis, we propose two applications utilizing garbling schemes. The first application provides privacy-preserving electronic surveillance. The second application extends electronic surveillance to more versatile monitoring, including also health telemetry. This kind of application would be ideal for assisted living services. In this work, we also present theoretical results related to garbling schemes. We present several new security definitions for garbling schemes which are of practical use. Traditionally, the same garbled algorithm can be evaluated once with garbled input. In applications, the same function is often evaluated several times with different inputs. Recently, a solution based on fully homomorphic encryption provides arbitrarily reusable garbling schemes. The disadvantage in this approach is that the arbitrary reuse cannot be efficiently implemented due to the inefficiency of fully homomorphic encryption. We propose an alternative approach. Instead of arbitrary reusability, the same garbled algorithm could be used a limited number of times. This gives us a set of new security classes for garbling schemes. We prove several relations between new and established security definitions. As a result, we obtain a complex hierarchy which can be represented as a product of three directed graphs. The three graphs in turn represent the different flavors of security: the security notion, the security model and the level of reusability. In addition to defining new security classes, we improve the definition of side-information function, which has a central role in defining the security of a garbling scheme. The information allowed to be leaked by the garbled algorithm and the garbled input depend on the representation of the algorithm. The established definition of side-information models the side-information of circuits perfectly but does not model side-information of Turing machines as well. The established model requires that the length of the argument, the length of the final result and the length of the function can be efficiently computable from the side-information function. Moreover, the side-information depends only on the function. In other words, the length of the argument, the length of the final result and the length of the function should only depend on the function. For circuits this is a natural requirement since the number of input wires tells the size of the argument, the number of output wires tells the size of the final result and the number of gates and wires tell the size of the function. On the other hand, the description of a Turing machine does not set any limitation to the size of the argument. Therefore, side-information that depends only on the function cannot provide information about the length of the argument. To tackle this problem, we extend the model of side-information so that side-information depends on both the function and the argument. The new model of side information allows us to define new security classes. We show that the old security classes are compatible with the new model of side-information. We also prove relations between the new security classes.Tämä väitöskirja käsittelee garblausskeemoja ja niiden sovelluksia. Garblausskeema on työkalu, jota käytetään turvallisen kahden osapuolen laskennan toteuttamiseen. Asiakas pitää hallussaan yksityistä algoritmia ja sen yksityistä syötettä, joilla hän haluaisi suorittaa tietyn laskennan. Asiakkaalla ei välttämättä ole riittävästi laskentatehoa, minkä vuoksi hän ei pysty suorittamaan laskentaa itse, vaan joutuu ulkoistamaan laskennan toiselle osapuolelle, palvelimelle. Koska asiakas tahtoo suojella algoritmiaan ja syötettään, hän ei voi vain lähettää niitä palvelimen laskettavaksi. Asiakas pystyy suojelemaan syötteensä ja algoritminsa yksityisyyttä käyttämällä garblausskeemaa. Verkkopohjaisten sovellusten kasvu on herättänyt huolta käyttäjien datan yksityisyyden turvasta. Siksi yksityisyyden säilyttävien tai yksityisyyden suojaa lisäävien tekniikoiden tutkimus on saanut huomiota. Garblaustekniikan avulla voidaan suojata sekä syöte että algoritmi. Lisäksi garblaukselle tiedetään olevan useita tehokkaita toteutuksia. Näiden syiden vuoksi garblausskeemat ovat houkutteleva tekniikka käytettäväksi yksityisyyden säilyttävien sovellusten toteutuksessa. Tässä työssä esittelemme kaksi sovellusta, jotka hyödyntävät garblaustekniikkaa. Näistä ensimmäinen on yksityisyyden säilyttävä sähköinen seuranta. Toinen sovellus laajentaa seurantaa monipuolisempaan monitorointiin, kuten terveyden kaukoseurantaan. Tästä voi olla hyötyä etenkin kotihoidon palveluille. Tässä työssä esitämme myös teoreettisia tuloksia garblausskeemoihin liittyen. Esitämme garblausskeemoille uusia turvallisuusmääritelmiä, joiden tarve kumpuaa käytännön sovelluksista. Perinteisen määritelmän mukaan samaa garblattua algoritmia voi käyttää vain yhdellä garblatulla syötteellä laskemiseen. Käytännössä kuitenkin samaa algoritmia käytetään usean eri syötteen evaluoimiseen. Hiljattain on esitetty tähän ongelmaan ratkaisu, joka perustuu täysin homomorfiseen salaukseen. Tämän ratkaisun ansiosta samaa garblattua algoritmia voi turvallisesti käyttää mielivaltaisen monta kertaa. Ratkaisun haittapuoli kuitenkin on, ettei sille ole tiedossa tehokasta toteutusta, sillä täysin homomorfiseen salaukseen ei ole vielä onnistuttu löytämään sellaista. Esitämme vaihtoehtoisen näkökulman: sen sijaan, että samaa garblattua algoritmia voisi käyttää mielivaltaisen monta kertaa, sitä voikin käyttää vain tietyn, ennalta rajatun määrän kertoja. Tämä näkökulman avulla voidaan määritellä lukuisia uusia turvallisuusluokkia. Todistamme useita relaatioita uusien ja vanhojen turvallisuusmääritelmien välillä. Relaatioiden avulla garblausskeemojen turvallisuusluokille saadaan muodostettua hierarkia, joka koostuu kolmesta komponentista. Tieto, joka paljastuu garblatusta algoritmista tai garblatusta syötteestä riippuu siitä, millaisessa muodossa algoritmi on esitetty, kutsutaan sivutiedoksi. Vakiintunut määritelmä mallintaa loogisen piiriin liittyvää sivutietoa täydellisesti, mutta ei yhtä hyvin Turingin koneeseen liittyvää sivutietoa. Tämä johtuu siitä, että jokainen yksittäinen looginen piiri asettaa syötteensä pituudelle rajan, mutta yksittäisellä Turingin koneella vastaavanlaista rajoitusta ei ole. Parannamme sivutiedon määritelmää, jolloin tämä ongelma poistuu. Uudenlaisen sivutiedon avulla voidaan määritellä uusia turvallisuusluokkia. Osoitamme, että vanhat turvallisuusluokat voidaan esittää uudenkin sivutiedon avulla. Todistamme myös relaatioita uusien luokkien välillä.Siirretty Doriast

Succinct Garbling Schemes and Applications

Assuming the existence of iO for P/poly and one-way functions, we show how to succinctly garble bounded-space computations (BSC) M: the size of the garbled program (as well as the time needed to generate the garbling) only depends on the size and space (including the input and output) complexity of M, but not its running time. The key conceptual insight behind this construction is a method for using iO to compress a computation that can be performed piecemeal, without revealing anything about it. As corollaries of our succinct garbling scheme, we demonstrate the following: -functional encryption for BSC from iO for P/poly and one-way functions; -reusable succinct garbling schemes for BSC from iO for P/poly and one-way functions; - succinct iO for BSC from sub-exponentially-secure iO for P/poly and sub-exponentially secure one-way functions; - (PerfectNIZK) SNARGS for bounded space and witness NP from sub-exponentially-secure iO for P/poly and sub-exponentially-secure one-way functions. Previously such primitives were only know to exists based on “knowledge-based” assumptions (such as SNARKs and/or differing-input obfuscation). We finally demonstrate the first (non-succinct) iO for RAM programs with bounded input and output lengths, that has poly-logarithmic overhead, based on the existence of sub-exponentially-secure iO for P/poly and sub-exponentially-secure one-way functions

Attribute-Based Encryption for Circuits of Unbounded Depth from Lattices: Garbled Circuits of Optimal Size, Laconic Functional Evaluation, and More

Although we have known about fully homomorphic encryption (FHE) from circular security assumptions for over a decade [Gentry, STOC \u2709; Brakerski–Vaikuntanathan, FOCS \u2711], there is still a significant gap in understanding related homomorphic primitives supporting all *unrestricted* polynomial-size computations. One prominent example is attribute-based encryption (ABE). The state-of-the-art constructions, relying on the hardness of learning with errors (LWE) [Gorbunov–Vaikuntanathan–Wee, STOC \u2713; Boneh et al., Eurocrypt \u2714], only accommodate circuits up to a *predetermined* depth, akin to leveled homomorphic encryption. In addition, their components (master public key, secret keys, and ciphertexts) have sizes polynomial in the maximum circuit depth. Even in the simpler setting where a single key is published (or a single circuit is involved), the depth dependency persists, showing up in constructions of 1-key ABE and related primitives, including laconic function evaluation (LFE), 1-key functional encryption (FE), and reusable garbling schemes. So far, the only approach of eliminating depth dependency relies on indistinguishability obfuscation. An interesting question that has remained open for over a decade is whether the circular security assumptions enabling FHE can similarly benefit ABE. In this work, we introduce new lattice-based techniques to overcome the depth-dependency limitations: - Relying on a circular security assumption, we construct LFE, 1-key FE, 1-key ABE, and reusable garbling schemes capable of evaluating circuits of unbounded depth and size. - Based on the *evasive circular* LWE assumption, a stronger variant of the recently proposed *evasive* LWE assumption [Wee, Eurocrypt \u2722; Tsabary, Crypto \u2722], we construct a full-fledged ABE scheme for circuits of unbounded depth and size. Our LFE, 1-key FE, and reusable garbling schemes achieve optimal succinctness (up to polynomial factors in the security parameter). Their ciphertexts and input encodings have sizes linear in the input length, while function digest, secret keys, and garbled circuits have constant sizes independent of circuit parameters (for Boolean outputs). In fact, this gives the first constant-size garbled circuits without relying on indistinguishability obfuscation. Our ABE schemes offer short components, with master public key and ciphertext sizes linear in the attribute length and secret key being constant-size

Interaction-Preserving Compilers for Secure Computation

In this work we consider the following question: What is the cost of security for multi-party protocols? Specifically, given an insecure protocol where parties exchange (in the worst case) ? bits in N rounds, is it possible to design a secure protocol with communication complexity close to ? and N rounds? We systematically study this problem in a variety of settings and we propose solutions based on the intractability of different cryptographic problems. For the case of two parties we design an interaction-preserving compiler where the number of bits exchanged in the secure protocol approaches ? and the number of rounds is exactly N, assuming the hardness of standard problems over lattices. For the more general multi-party case, we obtain the same result assuming either (i) an additional round of interaction or (ii) the existence of extractable witness encryption and succinct non-interactive arguments of knowledge. As a contribution of independent interest, we construct the first multi-key fully homomorphic encryption scheme with message-to-ciphertext ratio (i.e., rate) of 1 - o(1), assuming the hardness of the learning with errors (LWE) problem. We view our work as a support for the claim that, as far as interaction and communication are concerned, one does not need to pay a significant price for security in multi-party protocols

Bounded Functional Encryption for Turing Machines: Adaptive Security from General Assumptions

The recent work of Agrawal et al., [Crypto \u2721] and Goyal et al. [Eurocrypt \u2722] concurrently introduced the notion of dynamic bounded collusion security for functional encryption (FE) and showed a construction satisfying the notion from identity based encryption (IBE). Agrawal et al., [Crypto \u2721] further extended it to FE for Turing machines in non-adaptive simulation setting from the sub-exponential learining with errors assumption (LWE). Concurrently, the work of Goyal et al. [Asiacrypt \u2721] constructed attribute based encryption (ABE) for Turing machines achieving adaptive indistinguishability based security against bounded (static) collusions from IBE, in the random oracle model. In this work, we significantly improve the state of art for dynamic bounded collusion FE and ABE for Turing machines by achieving adaptive simulation style security from a broad class of assumptions, in the standard model. In more detail, we obtain the following results: - We construct an adaptively secure (AD-SIM) FE for Turing machines, supporting dynamic bounded collusion, from sub-exponential LWE. This improves the result of Agrawal et al. which achieved only non-adaptive (NA-SIM) security in the dynamic bounded collusion model. - Towards achieving the above goal, we construct a ciphertext policy FE scheme (CPFE) for circuits of unbounded size and depth, which achieves AD-SIM security in the dynamic bounded collusion model from IBE and laconic oblivious transfer (LOT). Both IBE and LOT can be instantiated from a large number of mild assumptions such as the computational Diffie-Hellman assumption, the factoring assumption, and polynomial LWE. - We construct an AD-SIM secure FE for Turing machines, supporting dynamic bounded collusions, from LOT, ABE for NC1 (or NC) and private information retrieval (PIR) schemes which satisfy certain properties. This significantly expands the class of assumptions on which AD-SIM secure FE for Turing machines can be based. In particular, it leads to new constructions of FE for Turing machines including one based on polynomial LWE and one based on the combination of the bilinear decisional Diffie-Hellman assumption and the decisional Diffie-Hellman assumption on some specific groups. In contrast the only prior construction by Agrawal et al. achieved only NASIM security and relied on sub-exponential LWE. To achieve the above result, we define the notion of CPFE for read only RAM programs and succinct FE for LOT, which may be of independent interest. - We also construct an ABE scheme for Turing machines which achieves AD-IND security in the standard model supporting dynamic bounded collusions. Our scheme is based on IBE and LOT. Previously, the only known candidate that achieved AD-IND security from IBE by Goyal et al. relied on the random oracle model